Distributed Control for Optimal Reactive Power Compensation in Smart Microgrid

Transcription

1 Distributed Control for Optimal Reactive Power Compensation in Smart Microgrid S. Bolognani and S. Zampieri Royal Institute of Technology, KTH October 14, 2011

2 1 Outline 2 Outline 3 Microgrid Model 4 Optimal Problem Model The Gradient of the Cost Function 5 The Proposed Optimization Algorithm Optimal Strategy 6 Simulations Conclusions

3 What is microgrid? A microgrid is a portion of the low-voltage power distributed network that is managed autonomously from the rest of the network, to achieve better quality of the service,... What is reactive power? The portion of power flow that is temporarily stored in the form of electric or magnetic fields, due to inductive and capacitive network elements, and returned to source Q = V rms I rms sin φ

4 1 Outline 2 Outline 3 Microgrid Model 4 Optimal Problem Model The Gradient of the Cost Function 5 The Proposed Optimization Algorithm Optimal Strategy 6 Simulations Conclusions Microgrid Model

5 Microgrid Model where: V is the set of nodes E is the set of edges G = (V, E, σ, τ), σ, τ : E V are two functions such that e E goes from the source node σ(e) to the terminal node τ(e) The incidence matrix of the graphy G is defined by 1 if v = σ(e) [A] ev = 1 if v = τ(e). 0 otherwise

6 Microgrid Model Outline Microgrid Model Here it limits the study to the steady state behavior of the system. u : V C is the voltage at the nodes i : V C is the current injected by the nodes ξ : E C is the current flowing on the edges A T ξ + i = 0, 1 T i = 0 (1) Au + Zξ = 0 (2) u(v)i(v) = s v, v V\{ v} (3) u( v) = u 0 (4)

7 Microgrid Model Cont. Microgrid Model s : V C, where s(v) := u(v)i(v) is the complex power injected by node v into the grid; f σ : E C, where f σ (e) := u(σ(e))ξ(e) is the power flow entering the edge e; f τ : E C, where f τ (e) := u(τ(e))ξ(e) is the power flow exiting the edge e. where Y = Z 1 A T f σ + s = 0 (5) f σ + Y u 0 (Au) = 0 (6)

8 Microgrid Model Cont. Microgrid Model Assumption 1: The inductance-resistance ratio is fixed for all the edges, i.e. z(e) = e jθ d(e), d(e) R +, e E Then we have Z = e jθ D, where D = diag(d(e), e E). Introduce weighted Laplacian matrix L = A T D 1 A, and the Green matrix X = L # [1]. We get the approximated node voltages û = ejθ u0 (Xs e T v Xs 1) + u 0 1 (7)

9 1 Outline 2 Outline Optimal Problem Model The Gradient of the Cost Function 3 Microgrid Model 4 Optimal Problem Model The Gradient of the Cost Function 5 The Proposed Optimization Algorithm Optimal Strategy 6 Simulations Conclusions

10 Optimal Problem Model The Gradient of the Cost Function The toal active power losses on the edges are given by ξ(e) 2 Rez(e) = cos θ(u ) T Lu e E Rel(e) = e E = cos θ u 0 2 (s ) T Xs = cos θ u 0 2 (pt Xp + q T Xq), (8) where p = Re(s) and q = Im(s), such that 1 T p = 0 and 1 T q = 0.

11 Optimal Problem Model The Gradient of the Cost Function Cont. Since we are allowed to command only a subset C V of the electronic interfaces connected to the microgrid. Furthermore, we assume that we are only allowed to command the amount of reactive power injected into the grid. min qc T Mq C + m T q C, (9) 1 T q C =c where M = cos θ u 0 2 X CC > 0, m = 2 cos θ u 0 2 X C C and c = 1T q C.

12 The Gradient of the Cost Function Optimal Problem Model The Gradient of the Cost Function The gradient of the cost function can be obtained by J(q C ) = 2Mq C + m. Then, the component of the gradient corresponding to a node v results to be [ J] v = 2 cos θ u 0 2 et v Xq 2 cos θ û 0 2 et v Im(e jθ û 0 u ) (10)

13 1 Outline 2 Outline The Proposed Optimization Algorithm Optimal Strategy 3 Microgrid Model 4 Optimal Problem Model The Gradient of the Cost Function 5 The Proposed Optimization Algorithm Optimal Strategy 6 Simulations Conclusions

14 The Proposed Optimization Algorithm Optimal Strategy In each set of the compensators, C i, nodes drives their state in a new feasible state that minimizes J(q C ) and keep those nodes s state, which are not in C i. The optimization subproblem faced by the nodes in C i can be rewritthen as min q T C i M Ci C i q Ci + (2q C i M C i C i + m T C i )q Ci (11a) s.t. 1 T q Ci = c 1 T q C i (11b) The optimal increment q Ci can be obtained by q Ci = M 1 C i C i 2 [ J] C i + 1T M 1 C i C i [ J] Ci 1 T M 1 C i C i 1 M 1 C i C i 2 1.

15 The Proposed Optimization Algorithm The Proposed Optimization Algorithm Optimal Strategy 1) a set C i is chosen according to a sequence of symbols η(t) {1,..., l}; 2) agents in C i sense the network and obtain an estimate of the gradient; 3) they determine a feasible update step that minimizes the given cost function; 4) they actuate the system by updating their state (the injected reactive power). q C (t + 1) = T η(t) [q C (t)] := argmin q C S η(t) J(q C (t) + q C ) (12)

16 The Proposed Optimization Algorithm Optimal Strategy Optimal Strategy: Nearest-neighbor Gossip It is can be proven that the bound β on the convergence rate of the algorithm satisfies β 1 1 N C 1 Proposition: Consider the nearest-neighbor clustering choice, corresponding to the set of the clusters {C e, e E}, where C e = {σ(e), τ(e)}. Assume that each set is triggered with the same probability. Then R β, with β = 1 1 N C 1

17 1 Outline 2 Outline Simulations Conclusions 3 Microgrid Model 4 Optimal Problem Model The Gradient of the Cost Function 5 The Proposed Optimization Algorithm Optimal Strategy 6 Simulations Conclusions

18 Simulations for Microgrid Model Simulations Conclusions Figure: Comparison between the network state (node voltages) computed via the exact model induced by Eq. (1) and (2) (circles), and the approximated model induced by Eq. (5) and (6) (stars). Figure: Contour plot of the exact distribution losses (thick line) and of the cost function whose gradient is given by the voltage measurements, according to Eq. (10) (thin line)

19 Simulations Conclusions Simualtions for the Optimal Algorithm Figure: Network: where compensators are in white, loads in gray. Figure: The algorithm behavior has been plotted for two different choices: nearest-neighbor gossip (solid line) and star topology (dashed). The dotted line represent the best possible performance.

20 Conculsions Outline Simulations Conclusions For the problem of optimal reactive power compensation in smart microgrid modelling the problem into the framework of quadratic optimization For the performance of the algorithm providing a bound on the best achievable performance giving the optimal performance which shows that the optimal strategy requires short-range communication

21 References I Outline Simulations Conclusions A. Ghosh, S. Boyd and A. Saberi,Minimizing effective resistance of a graph, SIAM Review, vol. 50, no. 1, pp , Feb